28278 + Interview Questions in Maths in Class 12 Page 87 InterviewSolution

4301.	If 1,omega, omega^2 are the three cube roots of unity, prove that (1+omega^2)^4=omega.
Answer» SOLUTION :L.H.S`=(1+OMEGA^2)^4=(-omega)^4=omega^4` `=omega^3.omega=1.omega=omega` ="R.H.S.(PROVED)"

Discussion

4302.	If int_(0)^(oo)e^(-ax)dx=(1)/(a), then value of int_(0)^(oo)x^(m)e^(-ax)dx equals
Answer» `((-1)^(m)m!)/(a^(m+1))` `((-1)^(m)(m-1))/(a^(m))` `(m!)/(a^(m))` `(m!)/(a^(m+1))` ANSWER :D

Discussion

4303.	Find all the points of discontinuity of the function f defined by f(x)= {(x+2",",ifx lt 1),(0, ifx =1),(x-2",", ifxgt 1):}
Answer» ANSWER :x=1

Discussion

4304.	Let A and D be opposite vertices of a square. A frog starts jumping at vertex A. From any vertex of the square except D, it may jump to either of the two adjacent vertices. When it reaches D, the from stop and stays there. The number of distinct path of exactly 6 jumps ending at D are ____________________
Answer» ANSWER :2

Discussion

4305.	Find the particular solution of the differentia equation : 2y e^(x//y)dx+(y - 2x e^(x//y))dx = 0, given that x = 0 when y = 1.
Answer» ANSWER :`2E^(X)/(y) + LOG \|y\| = 2`

Discussion

4306.	There are 7 distinguishable rings. The number of possible five - rings arrangements on the four fingers (except the thumb) of one hand (the order of the rings on each finger is to be counted and it is not required that each finger has a ring is equal to
Answer» 214110 211410 124110 141120 Answer :D

Discussion

4307.	The reflection of the point (alpha,beta,gamma) in the xy plane is
Answer» `(ALPHA,BETA,0)` `(0,0,GAMMA)` `(-alpha,-beta,gamma)` `(alpha,beta,-gamma)` ANSWER :D

Discussion

4308.	If A is a square matrix of order 5 and9A^(-1)then [adj (adj AJJ] (whereA^(-1) A^(T) and adj (A) denotes the inverse transpose and adjoint of matrix A respectively ) contains. (log 3=0.477,log 2=0.303)
Answer» ` 56` digits 60 digits 58 digits 53 digits ANSWER :A

Discussion

4309.	Evaluate the following integrals int (2x-3 cosx +e^x) dx
Answer» SOLUTION :`int(2x-3cosx+e^x)DX = 2x^2/2 -3(sinx) + e^x +c = x^2 - 3SINX +e^x +c`

Discussion

4310.	Consider the vectors oversetrarra=2overset^^i+3overset^^j+overset^^k "and" oversetrarrb=3overset^^i +2 overset^^j -3overset^^kfind oversetrarra.oversetrarrb
Answer» SOLUTION :`veca.vecb=2xx3+3xx2+mxx-3`

Discussion

4311.	(i) If \|z-3i lt sqrt5 then prove that the complex number z also satisfies the Inequality\|i(z+1)+1 \| lt 2 sqrt5. (ii) Find the complex number z which satisfies the condition \|z-a + ai\|=1 and has thegreatest absolute value where a is a real constant (a ne 0).
Answer» Answer :`\|z\|_("max") = (SQRT(a^2 + 4)+a)/(2), z= PM((sqrt(a^2 + 4) +a)/(2)) i and \|z\|_("min") = (sqrt(a^2 + 4)-a)/(2) , z= pm ((sqrt(a^2 + 4)-a)/(2))i`, this is true for all real `a GE 0; z = pm 2I`

Discussion

4312.	For each integer n ge 1, define a _(n) = [ (n)/( [sqrtn]) ], where [x] denotes the largest integer not exceeding x, for any real number x. Find the number of all n is the set {1,2,3,….,2010} for which a _(n)gt a _(n+1).
Answer» ANSWER :`2 ^(2)-1, 3 ^(2)-1,…..,44^(2)-1`

Discussion

4313.	Evaluate d/dx(a^(sinx)+e^-x) by chain rule.
Answer» SOLUTION :`d/dx(a^(SINX)+E^-X)d/dx(a^sin)d/dx(e^(-x))`

Discussion

4314.	f(x) = sqrtx, a = 1, b = 4 find c in Lagrange's mean value theorm:
Answer» `(9)/(4)` `(3)/(2)` `(1)/(2)` `(1)/(4)` Answer :A

Discussion

4315.	Using elementry transformation, find the inverse of the matrices. A = [(1,1/2),(2,3/2)]
Answer» SOLUTION :`A^(-1) = [(1,1/2),(2,3/2)]`

Discussion

4316.	Let N = 6 + 66 + 666 + ... + 666....66, where there are hundred 6's in the last term in the sum. How many times does the digit 7 occur in the number N ?
Answer» ANSWER :33

Discussion

4317.	Ifthere are 30 onto, mapping from a set containing n elements to theset {0,1} then n equals
Answer» 3 5 7 2 Answer :B

Discussion

4318.	g(x) = 1 + sqrt(x) and f(g(x)) =3 +2sqrtx+x then f(x) = ......
Answer» `1+2x^(2)` `2+x^(2)` `1+x` `2+x` SOLUTION :N/A

Discussion

4319.	Find the values of sin^(-1)("sin"(3pi)/5)
Answer» ANSWER :`(2PI)/5`

Discussion

4320.	Show that equation (a^2)/(x-a')+(b^2)/(x-b')+(c^3)/(x-c')+......+(k^2)/(x-k')=x-mWhere a ,b,c…….k, m,a',b',c'…….k' are allreal numbers , cannot have a non real root .
Answer» ANSWER :This is a CONTRADICTION . `therefore` The GIVENEQUATION cannot have NON -real roots .

Discussion

4321.	Find the number of distinct terms in the following expansions. (p+q)^(70)+(p-q)^(70) +(p+qi)^(70) +(p-qi)^(70) where i=sqrt(-1)
Answer» SOLUTION :N/A

Discussion

4322.	The solution of y dx - x dy + 3x^(2) y^(2) e^(x) dx = 0 is
Answer» `(y)/(X) + E^(x^(3)) = C` `(x)/(y) - e^(x^(3)) = c` `(x)/(y) + e^(x^(3)) =c` `(y)/(x) - e^(x^(3)) = c^(3)` ANSWER :C

Discussion

4323.	If [vec(a)xx vec(b),vec(b)xx vec( c ),vec( c )xx vec(a)]=lambda[vec(a),vec(b),vec( c )]^(2) then lambda = …………..
Answer» 0 1 2 3 Answer :B

Discussion

4324.	If f is defined in [1, 3] by f(x)=x^3+bx^2+ax such that f(1) - f(3) = 0 and f'(c) =0 where c=2+1/root()3, then (a, b) =
Answer» `(-6,11)` `(2-1/root()3,2+1/root()3)` `(11,-6)` `(6,11)` ANSWER :C

Discussion

4325.	The mean and standard deviation of 10 items were found to be 17 and sqrt(33). Later it was detected that an item was tekn wrongly as 26 in place of 12. Find the correct mean and standard deviation.
Answer» ANSWER :MEAN = 15.6 and s.d = 5.04

Discussion

4326.	P is a point on the hyperbolaThe tangent at P meets the transverse axis at T, N is the foot of the perpendicular from P to the transverse axis. If O is the origin, then ON.OT is equal to.
Answer» 81 49 `81` `-49` ANSWER :A

Discussion

4327.	A particle is displaced from the point whose position vector is 5hati+ hatj + hatk to the point 9hati + 3hatj + 8hatk under the action of constant forces difined by 9hati + 5hatj + hatk, 6hati- 3hatj - 2hatk and 7hati -8hatj+ 3hatk. The work done by these forces is
Answer» 0 24 32 92 Answer :D

Discussion

4328.	Evaluate int (3x-2) sqrt(2x^(2)-x+1) dx
Answer» Answer :`=(1)/(2)(2x^(2)-X+1)^(3//2)-(5)/(64sqrt(2))=(x-(1)/(4)) sqrt((1-(1)/(4))^(2)+(7)/(16)))-(35)/(64sqrt(2)) SINH^(-1)((4x-1)/(sqrt(7)))+C`

Discussion

4329.	Range of the function f(x)=sqrt(abs(sin^(-1)abs(sinx))cos^(-1)abs(cosx)) is
Answer» {0} `[0,SQRT(PI/2)]` `[0,sqrt(pi)]` NONE of these Answer :A

Discussion

4330.	Evaluate the following integrals. int(1)/(8+2x^(2))dx
Answer» ANSWER :`(1)/(4)TAN^(-1)((X)/(2))+c`

Discussion

4331.	Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Answer» ANSWER :X=6, 4, 2, 0

Discussion

4332.	A die is rolled until a 6 is obtained. What is the probability that you end up in the second roll
Answer» Solution :A die rolled until a 6 is obtained We are to end up in the 2nd ROLL i.e., we GET 6 in the 2nd roll. Let A be the EVENT of getting a 6 in one roll of a dia. `therefore P(A)=1/6 implies P(A.)=1-1/6=5/6` `therefore` PROBABILITY of getting a 6 in the 2nd roll `=5/6xx1/6=5/35`

Discussion

4333.	Let {:(l rarr 0),(lim):}(1+5t)^(1/t)=K and X be the random variable representingnumber of successes in 100 independent trials. If the probability of success in each trial is 0.05, then the probability of getting at least one success is
Answer» `(1-K)/(K)` `(K-1)/(K)` `(K+1)/(2K)` `(5K+2)/(7K)` ANSWER :B

Discussion

4334.	Find the points on the curve y=x^(3) at which the slope of the tangent is equal to the y - coordinate of the point.
Answer» ANSWER :(0, 0) and (3, 27)

Discussion

4335.	Compute the integral int_(0)^(pi^(2)/4) sin sqrt(x)dx
Answer» ANSWER :2

Discussion

4336.	Find the equation of the chord joining two points (x_(1), y_(1))and (x_(2), y_(2)) on t he rectangularhyperbola xy = c^(2)
Answer» ANSWER :X/(x1+x2) + y/(y1+y2) =1

Discussion

4337.	consider the function f(x) =1(1+(1)/(x))^(x) The function f(x)
Answer» has a maximum but no minima has a minima but no MAXIMA has exactly ONE maxima and one minima is monotnic Solution :`f(x)=1+(1)/(X)^(X)` f(x) is defined if `1+1/xgt0 or (x+1)/(x)gt0` or `(-oo,-1)cup(0,oo)` Now f(X)=`(1+(1)/(x))^(x)ln (1+(1)/(x))+(x)/(1+(1)/(x))(-1)/(x^(2))` `=(1+(1)/(x))^(x)ln(1+(1)/(x))-(1)/(x+1)` Now `(1+(1)/(x))^(x)` is always positive Hence the sign of f(X) DEPENDS on the sign of ln `(1+(1)/(x))-(1)/(1+x)` Let `g(x)=ln 1+(1)/(x)-(1)/(x+1)` `g(x)=(1)/1+(1)/(x)(-1)/(x^(2))+(1)/(x+1)^(2)=(-1)/(x(x+1)^(2)` (i)for `x in (0,oo)g(x)lt0` Thus g(X) is monotonically decreasing for x in `(0,oo)` or `g(X) gt underset(xrarroo)limg(x)` or `g(X)gt0, so f(X) gt0` (ii)for `x in (-oo,-1),g(x)gt0` Thus g(X) is monotonically increasing for `x in (-oo,-1)` or `g(X) gt underset(xrarroo)limg(X)gt0` `therefore f(x)gt0` Hence form (i) nd (ii) we GET `f(X) gt0 forall x in (-oo,-1)cup(0,oo)` Thus f(x) is montonically increasing in its domain Also `underset(xrarroo)LIM(1+(1)/(x))^(x)`=e `underset(xrarr0)lim(1+(1)/(x))^(x)=1 and underset(xrarr-1)lim(1+(1)/(x))^(x)=oo` The graph of f(X) is shown in figure Range is `y in (1,oo)-{e}`

Discussion

4338.	Find the number of ways of arranging the letters of the word 'BRINGING' so that they begin and end with I.
Answer» ANSWER :180

Discussion

4339.	Find the equation of the circle with centre C and radius r where C = (-a,-b) , r = sqrt(a^(2) -b^(2)) (\|a\|\|gt\|b\|)
Answer» Answer :`X ^(2) +y^(2)+2ax +2by+ 2B^(2) =0 `

Discussion

4340.	If the plane 2x + 3y + 4z = 1 intersects X-axis, Y-axis and Z-axisat the points A,B and C repectively, then the centroid of a Delta ABC is................
Answer» `(2/3,1,4/3)` `(6,9,12)` `(1/6,1/9,1/12)` `(1/2 ,1/3,1/4)` ANSWER :C

Discussion

4341.	Out of 7 tickets consecutively numbered 3 are drawn at random. The probabilityfor the numbers on the ticketsto be in A.P. is
Answer» `(9)/(35)` `(""^(5)C_(3))/(""^(7)C_(3))` `(10)/(35)` `(2)/(5)` Answer :A

Discussion

4342.	Find the asymptotes of the following curves : y = "In " (4-x^(2))
Answer» ANSWER :`X = PM 2`

Discussion

4343.	Locus of the mid-point of the chord of the hyperbolawhich is a tangent to the circle x^(2) +y^(2) = c is
Answer» `((x^(2))/(a^(2))-(y^(2))/(B^(2)))=C^(2)((x^(2))/(a^(4))+(y^(2))/(b^(4)))` `((x^(2))/(a^(2))-(y^(2))/(b^(2)))=c^(2)((x^(2))/(a^(4))-(y^(2))/(b^(4)))` `((x^(2))/(a^(2))-(y^(2))/(b^(2)))=c^(2)((x^(2))/(a^(4))+(y^(2))/(b^(2)))` `((x^(2))/(a^(4))-(y^(2))/(b^(4)))=c^(2)((x^(2))/(a^(4))+(y^(2))/(b^(4)))` ANSWER :a

Discussion

4344.	The smaller area of the region cut off by x+y=2 from the circle x^(2)+y^(2)=4 is
Answer» `pi-2` `pi-4` `2pi-4` `4pi-2` ANSWER :A

Discussion

4345.	Find the absolute maximum and minimum values of the function given byf(x)=cos^(2)x+sin x, x in [0, pi]
Answer» ANSWER :`(5)/(4)`, 1

Discussion

4346.	Function f(x) = \|sinx\|, x in (- (pi)/(2) , 0) is :
Answer» Only an INCREASING Neither increasing nor decreasing Strictly increasing Strictly decreasing Answer :D

Discussion

4347.	Given that veca.vecb=0 and veca xx vecb=0. What can you conclude about the vectors veca and vecb.
Answer» `VECA BOT VECB` `veca\|\|vecb` `veca=0` EITHER `\|veca\|=0 or \|vecb\|=0` SOLUTION :N/A

Discussion

4348.	Determine order and Degree(if defined) of differential equations given ((ds)/(dt))^(4) + 3s(d^(2)s)/(dt^(2)) = 0
Answer» ANSWER :ORDER 2; DEGREE 1

Discussion

4349.	Find maximum and minimum value of the following functions in the given interval.f(x)=xe^(-x), x in [0, oo]
Answer» Answer :MAXIMUM VALUE `(1)/(e )` Minimum value 0

Discussion

4350.	If the complex numbers z_(1) , z_(2) , z_(3) represents the vertices of an equilateral triangle such that \|z_1\| = \|z_2\| = \|z_3\| then
Answer» `z_(1) + z_(2) = z_(3)` `z_(1) + z_(3) = z_(2)` `z_(2) + z_(3) = z_(1)` `z_(1) + z_(2) + z_(3) = 0` Answer :D

Discussion

Explore topic-wise InterviewSolutions in .

If 1,omega, omega^2 are the three cube roots of unity, prove that (1+omega^2)^4=omega.

If int_(0)^(oo)e^(-ax)dx=(1)/(a), then value of int_(0)^(oo)x^(m)e^(-ax)dx equals

Find all the points of discontinuity of the function f defined by f(x)= {(x+2",",ifx lt 1),(0, ifx =1),(x-2",", ifxgt 1):}

Find the particular solution of the differentia equation : 2y e^(x//y)dx+(y - 2x e^(x//y))dx = 0, given that x = 0 when y = 1.

There are 7 distinguishable rings. The number of possible five - rings arrangements on the four fingers (except the thumb) of one hand (the order of the rings on each finger is to be counted and it is not required that each finger has a ring is equal to

The reflection of the point (alpha,beta,gamma) in the xy plane is

If A is a square matrix of order 5 and9A^(-1)then [adj (adj AJJ] (whereA^(-1) A^(T) and adj (A) denotes the inverse transpose and adjoint of matrix A respectively ) contains. (log 3=0.477,log 2=0.303)

Evaluate the following integrals int (2x-3 cosx +e^x) dx

Consider the vectors oversetrarra=2overset^^i+3overset^^j+overset^^k "and" oversetrarrb=3overset^^i +2 overset^^j -3overset^^kfind oversetrarra.oversetrarrb

(i) If |z-3i lt sqrt5 then prove that the complex number z also satisfies the Inequality|i(z+1)+1 | lt 2 sqrt5. (ii) Find the complex number z which satisfies the condition |z-a + ai|=1 and has thegreatest absolute value where a is a real constant (a ne 0).

For each integer n ge 1, define a _(n) = [ (n)/( [sqrtn]) ], where [x] denotes the largest integer not exceeding x, for any real number x. Find the number of all n is the set {1,2,3,….,2010} for which a _(n)gt a _(n+1).

Evaluate d/dx(a^(sinx)+e^-x) by chain rule.

f(x) = sqrtx, a = 1, b = 4 find c in Lagrange's mean value theorm:

Using elementry transformation, find the inverse of the matrices. A = [(1,1/2),(2,3/2)]

Let N = 6 + 66 + 666 + ... + 666....66, where there are hundred 6's in the last term in the sum. How many times does the digit 7 occur in the number N ?

Ifthere are 30 onto, mapping from a set containing n elements to theset {0,1} then n equals

g(x) = 1 + sqrt(x) and f(g(x)) =3 +2sqrtx+x then f(x) = ......

Find the values of sin^(-1)("sin"(3pi)/5)

Show that equation (a^2)/(x-a')+(b^2)/(x-b')+(c^3)/(x-c')+......+(k^2)/(x-k')=x-mWhere a ,b,c…….k, m,a',b',c'…….k' are allreal numbers , cannot have a non real root .

Find the number of distinct terms in the following expansions. (p+q)^(70)+(p-q)^(70) +(p+qi)^(70) +(p-qi)^(70) where i=sqrt(-1)

The solution of y dx - x dy + 3x^(2) y^(2) e^(x) dx = 0 is

If [vec(a)xx vec(b),vec(b)xx vec( c ),vec( c )xx vec(a)]=lambda[vec(a),vec(b),vec( c )]^(2) then lambda = …………..

If f is defined in [1, 3] by f(x)=x^3+bx^2+ax such that f(1) - f(3) = 0 and f'(c) =0 where c=2+1/root()3, then (a, b) =

The mean and standard deviation of 10 items were found to be 17 and sqrt(33). Later it was detected that an item was tekn wrongly as 26 in place of 12. Find the correct mean and standard deviation.

P is a point on the hyperbolaThe tangent at P meets the transverse axis at T, N is the foot of the perpendicular from P to the transverse axis. If O is the origin, then ON.OT is equal to.

A particle is displaced from the point whose position vector is 5hati+ hatj + hatk to the point 9hati + 3hatj + 8hatk under the action of constant forces difined by 9hati + 5hatj + hatk, 6hati- 3hatj - 2hatk and 7hati -8hatj+ 3hatk. The work done by these forces is

Evaluate int (3x-2) sqrt(2x^(2)-x+1) dx

Range of the function f(x)=sqrt(abs(sin^(-1)abs(sinx))cos^(-1)abs(cosx)) is

Evaluate the following integrals. int(1)/(8+2x^(2))dx

Let X represent the difference between the number of heads and the number of﻿ tails obtained when a coin is tossed 6 times. What are possible values of X?﻿

A die is rolled until a 6 is obtained. What is the probability that you end up in the second roll

Let {:(l rarr 0),(lim):}(1+5t)^(1/t)=K and X be the random variable representingnumber of successes in 100 independent trials. If the probability of success in each trial is 0.05, then the probability of getting at least one success is

Find the points on the curve y=x^(3) at which the slope of the tangent is equal to the y - coordinate of the point.

Compute the integral int_(0)^(pi^(2)/4) sin sqrt(x)dx

Find the equation of the chord joining two points (x_(1), y_(1))and (x_(2), y_(2)) on t he rectangularhyperbola xy = c^(2)

consider the function f(x) =1(1+(1)/(x))^(x) The function f(x)

Find the number of ways of arranging the letters of the word 'BRINGING' so that they begin and end with I.

Find the equation of the circle with centre C and radius r where C = (-a,-b) , r = sqrt(a^(2) -b^(2)) (|a||gt|b|)

If the plane 2x + 3y + 4z = 1 intersects X-axis, Y-axis and Z-axisat the points A,B and C repectively, then the centroid of a Delta ABC is................

Out of 7 tickets consecutively numbered 3 are drawn at random. The probabilityfor the numbers on the ticketsto be in A.P. is

Find the asymptotes of the following curves : y = "In " (4-x^(2))

Locus of the mid-point of the chord of the hyperbolawhich is a tangent to the circle x^(2) +y^(2) = c is

The smaller area of the region cut off by x+y=2 from the circle x^(2)+y^(2)=4 is

Find the absolute maximum and minimum values of the function given byf(x)=cos^(2)x+sin x, x in [0, pi]

Function f(x) = |sinx|, x in (- (pi)/(2) , 0) is :

Given that veca.vecb=0 and veca xx vecb=0. What can you conclude about the vectors veca and vecb.

Determine order and Degree(if defined) of differential equations given ((ds)/(dt))^(4) + 3s(d^(2)s)/(dt^(2)) = 0

Find maximum and minimum value of the following functions in the given interval.f(x)=xe^(-x), x in [0, oo]

If the complex numbers z_(1) , z_(2) , z_(3) represents the vertices of an equilateral triangle such that |z_1| = |z_2| = |z_3| then

Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?